Question: A new shopping mall is gaining in popularity. Every day since it opened, the number of shoppers is 20% more than the number of shoppers the day before. The total number of shoppers over the first 4 days is 671.
How many shoppers were at the mall on the first day?
Round your final answer to the nearest integer.
1. **State the problem:** We want to find the number of shoppers on the first day given that each day the number of shoppers increases by 20% from the previous day, and the total number of shoppers over 4 days is 671.
2. **Define variables and formula:** Let the number of shoppers on the first day be $x$.
Each subsequent day has 20% more shoppers, so the number of shoppers on day 2 is $x \times 1.2$, on day 3 is $x \times 1.2^2$, and on day 4 is $x \times 1.2^3$.
The total number of shoppers over 4 days is:
$$x + x \times 1.2 + x \times 1.2^2 + x \times 1.2^3 = 671$$
3. **Simplify the equation:** Factor out $x$:
$$x(1 + 1.2 + 1.2^2 + 1.2^3) = 671$$
Calculate the sum inside the parentheses:
$$1 + 1.2 + 1.44 + 1.728 = 5.368$$
So the equation becomes:
$$x \times 5.368 = 671$$
4. **Solve for $x$:**
$$x = \frac{671}{5.368}$$
Show cancellation step:
$$x = \frac{\cancel{671}}{\cancel{5.368}}$$
Calculate the division:
$$x \approx 124.99$$
5. **Round the answer:** The number of shoppers on the first day is approximately $125$.
**Final answer:**
$$\boxed{125}$$